Title: Subnormalized states and trace-nonincreasing maps

Abstract

We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M{sub N} of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt (HS) (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace-nonincreasing completely positive maps belong to the N cube inscribed in the set of subnormalized states of size N. As a by-product we derive the measure in M{sub N} induced by partial trace of mixed quantum states distributed uniformly with respect to the HS measure in M{sub N{sup 2}}.

It is shown that for arbitrary {ital n}, there exists a trace map for any {ital n}-letter substitutional sequence. Trace maps are explicitly obtained for the well-known circle and Rudin-Shapiro sequences which can be defined by means of substitution rules on three and four letters, respectively. The properties of the two trace maps and their consequences for various spectral properties are briefly discussed.

We address the problem of finding optimal CPTP (completely positive trace-preserving) maps between a set of binary pure states and another set of binary generic mixed state in a two-dimensional space. The necessary and sufficient conditions for the existence of such CPTP maps can be discussed within a simple geometrical picture. We exploit this analysis to show the existence of an optimal quantum repeater which is superior to the known repeating strategies for a set of coherent states sent through a lossy quantum channel. We also show that the geometrical formulation of the CPTP mapping conditions can be a simplermore » method to derive a state-dependent quantum (anti) cloning machine than the study so far based on the explicit solution of several constraints imposed by unitarity in an extended Hilbert space.« less

The ability of fully reconstructing quantum maps is a fundamental task of quantum information, in particular when coupling with the environment and experimental imperfections of devices are taken into account. In this context, we carry out a quantum process tomography approach for a set of non-trace-preserving maps. We introduce an operator P to characterize the state-dependent probability of success for the process under investigation. We also evaluate the result of approximating the process with a trace-preserving one.

A summary of exploration and development drilling in Illinois, Indiana, Kentucky, and Tennessee is presented. Tennessee and Indiana reported increases in crude oil production, while Illinois and Kentucky reported decreases. Exploration activity was down in all these states in 1979, except Tennessee, which showed a small increase. Data are presented in a number of tables, and maps of the states show the most important areas of exploration and development.

Measured surface radium content, geologic province information, information on the fraction of homes with basements and with living-area basements, and measurements from the EPA/State Residential Radon Surveys, were used in a Bayesian mixed effects regression to predict the distributions of short-term winter and annual living-area average radon concentrations by county in the mid-Atlantic states. The information provided by those explanatory variables is roughly equivalent to collecting an extra 12 observations per county, effectively doubling the amount of information in a typical county. Predicted county geometric means are subject to standard errors of 15 % to 30 % for typical counties,more » with the Uncertainty in a given county depending on the number of radon measurements in the county and the amount of information about the geologic province that contains the county. After controlling for soil radium concentration and the effect of measuring in a basement vs. the first floor, typical geologic provinces are found to be associated with elevation or depression of indoor radon concentrations by 30% on average, with some provinces having effects of considerably larger magnitude. 20 refs., 7 figs., 3 tabs.« less